Lack of superstable trajectories in linear viscoelasticity: a numerical approach

Given a positive operator A on some Hilbert space, and a nonnegative decreasing summable function μ , we consider the abstract equation with memory u ¨ ( t ) + A u ( t ) - ∫ 0 t μ ( s ) A u ( t - s ) d s = 0 modeling the dynamics of linearly viscoelastic solids. The purpose of this work is to provide numerical evidence of the fact that the energy E ( t ) = ( 1 - ∫ 0 t μ ( s ) d s ) ‖ u ( t ) ‖ 1 2 + ‖ u ˙ ( t ) ‖ 2 + ∫ 0 t μ ( s ) ‖ u ( t ) - u ( t - s ) ‖ 1 2 d s of any nontrivial solution cannot decay faster than exponential, no matter how fast might be the decay of the memory kernel μ . This will be accomplished by simulating the integro-differential equation for different choices of the memory kernel μ and of the initial data.